We recap the classical CG error bound...
which can be solved using the Chebyshev polynomial...
actually, a scaled Chebyshev polynomial...
But what if we reintroduce the full spectrum of A?
we consider the simplest, non-trivial case of two-cluster spectra.
How can we solve the min-max problem in this case?
We use the product of two scaled Chebyshev polynomials.
This leads to the two-cluster CG iteration bound from Axelsson (1976).
We actually want to develop the most general bound...
leading to the most general CG iteration bound.
This is a simplified view of the general CG iteration bound.
But how do we get from the spectrum of A to a set of clusters?
We start by assuming no knowledge of the cluster boundaries...
we find the largest relative gap in the spectrum.
If this condition is satisfied, we perform recursion on the two partitions.
This results in two partitions.
This results in two partitions.
This results in two partitions.
This results in two partitions.